Tagged Questions
2
votes
1answer
31 views
Calculation Of Integral Related To Sequence
Let's evaluate the following integral. Many trials but no success.
$$\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$$
4
votes
3answers
56 views
Integration using summation
How do you integrate $\sqrt{x}$ from an arbitrary constant $a$ to another $b$ by summation ?
6
votes
1answer
77 views
integration as limit of a sum
If $f$ is continuous on $[0, 1]$ then
$$\lim_ {n\to\infty}\sum_{j=0}^{\lfloor n/2\rfloor} \frac1{n}f\left(\frac {j}{n}\right) = ? $$
will the answer be that the limit exists and is equal to $ ...
1
vote
1answer
95 views
Proof that Riemann-integrability is preserved by changing the function at a converging sequence of points
Let $(x_n)^{\infty}_{n=1}\in[a,b]$ such that $\lim\limits_{n\to\infty}x_n=l$. Let $g:[a,b]\to \mathbb{R}$ be bounded function. Assume that there exists a Riemann integrable function $f:[a,b]\to ...
9
votes
1answer
96 views
Evaluating a slow sum
In my integration adventures, I came across this sum which I could not simplify:
$$\sum_{n=1}^{\infty}\frac{(-1)^{n}\log(2n+1)}{2n+1}$$
Wolfram seems to believe the sum diverges and is not of much ...
9
votes
2answers
112 views
How to evaluate $\displaystyle\int_0^1\frac{\log^2(1+x)}x\mathrm dx$?
The definite integral
$$\int_0^1\frac{\log^2(1+x)}x\mathrm dx=\frac{\zeta(3)}4$$
arose in my answer to this question. I couldn't find it treated anywhere online. I eventually found two ways to ...
8
votes
4answers
195 views
Calculate : $\int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x $
Find : $\displaystyle \int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x $.
I've done some work but I've got stuck, you may try to help me continue or give me another way , in both cases ...
1
vote
3answers
57 views
A rational function integral
How to evaluate :
$$\int_{0}^{1}{\frac{{{x}^{a-1}}}{1+{{x}^{b}}}}\text{d}x,\ \ \ a,\ b\in {{\mathbb{N}}^{+}}$$
1
vote
2answers
73 views
A integral with hyperbolic function
Evaluate :
$$\int_{0}^{2{\rm arccosh\,} \pi }{\frac{\text{d}x}{1+\frac{{{\sinh }^{2}}x}{{{\pi }^{4}}}}}$$
0
votes
2answers
86 views
How to calculate a infinite Riemann sum
I am working on this assignment and I got a little stuck up with this. I got some ideas but I not at all sure if they are correct. So I am hoping to get opinion from someone in here please.
How to ...
1
vote
1answer
279 views
Another two hard integrals
Evaluate :
$$\begin{align}
& \int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\left( 2\cos x \right)}{{{\ln }^{2}}\left( 2\cos x \right)+{{x}^{2}}}}\text{d}x \\
& \int_{0}^{1}{\frac{\arctan ...
1
vote
0answers
42 views
Alternative explanation for $\iint_D \left|\log \left( \frac{e}{1-z} \right) \right|^2 \ dA = \frac{\pi^3}{6}$?
I thought up a curious definite integral. Let $D = \{ z \in \mathbb{C} : |z|<1\}$. Let $A$ denote area measure on $D$, normalized so that $A(D) = \pi$. I claim that $$\iint_D \left|\log \left( ...
5
votes
2answers
103 views
Prove that $\int_{0}^{1} \ln\left(\frac{1-a x}{1-a}\right) \frac{1}{\ln x} \mathrm{dx} = -\sum_{k=1}^{\infty} a^{k} \frac{\ln(1+k)}{k}, \space a<1$
Prove that
$$\int_{0}^{1} \ln\left(\frac{1-a x}{1-a}\right) \frac{1}{\ln x} \mathrm{dx} = -\sum_{k=1}^{\infty} a^{k} \frac{\ln(1+k)}{k}, \space a<1$$
I find this question rather troublesome since ...
11
votes
3answers
307 views
Compute $\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$
Compute the limit
$$\lim_{n\to\infty} \left(\int_0^{\pi} \frac{\sin^2 n x}{\sin x} \ dx-\sum_{k=1}^n \frac{1}{k}\right)$$
0
votes
2answers
69 views
Show that the integral of this rational function is equal to an infinite alternating harmonic series
One of my friends gave me the following question from his review, I have little experience to dealing with these types of questions in Analysis so if you could help us just to get started it would be ...
5
votes
1answer
195 views
expansion of $\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt$ in inverse powers of $p$
This question relates to this answer I gave to a question about the integral
$$\int_0^\infty\left(\frac{\sin t}t\right)^p\mathrm dt\;.$$
I derived an expansion in inverse powers of $p$ and then ...
1
vote
1answer
213 views
Advanced application of the Binomial Theorem
I'm trying to solve the following integral:
$$
\int_{-1}^{1}C_{n_1-l_1}^{l_1+1}(x)C_{n_2-l_2}^{l_2+1}(x)C_{n_3-l_3}^{l_3+1}(x)(1-x^2)^{(l_1+l_2+l_3+1)/2}dx
$$
Where $C_{n}^{\lambda}(x)$ is a ...
2
votes
1answer
97 views
Convergence of a characteristic function
This is the last part of a three part problem on characteristic functions, and it's been driving me crazy over the last few days. Any help would be most appreciated.
$X_1,X_2, \ldots, X_n$ are ...
